Finitely presentable morphisms in exact sequences
Mathematics & Actuarial Science Department
Theory and Applications of Categories
Let K be a locally finitely presentable category. If K is abelian and the sequence, we show that 1) K is finitely generated â‡” c is finitely presentable; 2) k is finitely presentable â‡” C is finitely presentable. The "â‡" directions fail for semi-abelian varieties. We show that all but (possibly) 2)(â‡) follow from analogous properties which hold in all locally finitely presentable categories. As for 2)(â‡), it holds as soon as K is also co-homological, and all its strong epimorphisms are regular. Finally, locally finitely coherent (resp. noetherian) abelian categories are characterized as those for which all finitely presentable morphisms have finitely generated (resp. presentable) kernel objects. Â© Michel HÃ©bert, 2010.
(2010). Finitely presentable morphisms in exact sequences. Theory and Applications of Categories, 24, 209–220.
"Finitely presentable morphisms in exact sequences." Theory and Applications of Categories, vol. 24, 2010, pp. 209–220.